It seems to me that as the respected head of a school, Mrs. Goddard could be expected to maintain confidentiality on such matters as a student's questionable parentage, and I'm sure Emma is a bit frustrated by that.
Also, it has always struck me that Emma has no idea what life must be like for Harriet, or indeed anyone of a lower class. Emma is proud of her family and has a strong sense of her social standing. Even with her lively imagination, I don't think Emma can begin to understand what it would be like not to know who her father was. Emma is a bit amused by the way Harriet speaks about the Martins, but I suspect that her stay with the Martins was the first glimpse the poor girl ever had of a happy family life.